N° 8 Espressioni Algebriche

$1 - \frac{x - 1}{x + 1} + \frac{2 x - x^{2} - 1}{1 + 2 x + x^{2}} + \frac{3 + 3 x + x^{2} + x^{3}}{1 + 3 x + 3 x^{2} + x^{3}} =$

$= 1 - \frac{x - 1}{x + 1} - \frac{{(x - 1)}^{2}}{{(x + 1)}^{2}} + \frac{(x + 1) (x^{2} + 3)}{{(x + 1)}^{3}} =$

$= 1 - \frac{x - 1}{x + 1} - \frac{{(x - 1)}^{2}}{{(x + 1)}^{2}} + \frac{x^{2} + 3}{{(x + 1)}^{2}} =$

$= \frac{{(x + 1)}^{2} - (x + 1) (x - 1) - {(x - 1)}^{2} + x^{2} + 3}{{(x + 1)}^{2}} =$

$= \frac{x^{2} + 2 x + 1 - x^{2} + 1 - x^{2} + 2 x - 1 + x^{2} + 3}{{(x + 1)}^{2}} =$

$= \frac{(1 - 1 - 1 + 1) x^{2} + (2 + 2) x + (1 + 1 - 1 + 3)}{{(x + 1)}^{2}} =$

$= \frac{4 x + 4}{{(x + 1)}^{2}} = \frac{4 (x + 1)}{{(x + 1)}^{2}} = \frac{4}{x + 1}$